Keywords

2.1 Introduction

The EEM [1,2,3,4], ER [5,6,7,8], EP [9,10,11,12], FE [13,14,15,16] and TP [17,18,19,20] have extensively been investigated in the recent literature, and they have important contributions in controlling control the transport phenomena in biosensing materials. Although biosensing materials find wide applications and many physical properties have already been studied [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37], nevertheless it appears from the literature that the study of the said electronic properties has yet to be made. In this chapter, they are being investigated in HDNWs of biosensing materials. It may be noted that HDNWs are also being studied by various workers [38,39,40]. The theoretical background is described in Sects. 2.2, and 2.3 contains the results and discussion in this context.

2.2 Theoretical Background

The \(E - k_{x}\) relation assumes the form [37]

$$ k_{x}^{2} = A_{11} (E,\eta_{g} ,n_{y} ) $$
(2.1)

where

$$ \begin{aligned} & A_{11} (E,\eta_{g} ,n_{y} ) \\ & = \left[ {\frac{2}{\sqrt 3 }\cos^{ - 1} \left[ {{{\left[ {[f\gamma (E,\eta_{g} ) + g]^{2} - 3 - D - 2\cos \left( {\frac{{n_{y} \pi }}{{d_{y} }}} \right)} \right]} \mathord{\left/ {\vphantom {{\left[ {[f\gamma (E,\eta_{g} ) + g]^{2} - 3 - D - 2\cos \left( {\frac{{n_{y} \pi }}{{d_{y} }}} \right)} \right]} {\left( {4\cos \left( {\frac{{3\pi n_{y} }}{{2d_{y} }}} \right)} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {4\cos \left( {\frac{{3\pi n_{y} }}{{2d_{y} }}} \right)} \right)}}} \right]} \right]^{2} \\ \end{aligned} $$

and the other notations are defined in [37]

The use of (2.1) leads to the expression of EEM as

$$ m^{*} (E_{F} ,\eta_{g} ,n_{y} ) = \frac{{\hbar^{2} }}{2}A^{\prime}_{11} (E_{F} ,\eta_{g} ,n_{y} ) $$
(2.2)

where the notations have their usual significances.

The \(n_{0}\) can be written as

$$ \begin{aligned} n_{y} & = \frac{{2g_{v} }}{\pi }\sum\limits_{{n_{y} = 1}}^{{n_{{y_{\max } }} }} {\left[ {\sqrt {A_{11} (E_{F} ,\eta_{g} ,n_{y} )} } \right.} \\ & \quad + \left. {\sum\limits_{r = 1}^{r = n} {2(1 - 2^{1 - 2r} )\xi (2r)\frac{{\partial^{2r} }}{{\partial E_{F}^{2r} }}\left[ {\sqrt {A_{11} (E_{F} ,\eta_{g} ,n_{y} )} } \right]} } \right] \\ \end{aligned} $$
(2.3)

where the notations have their usual significances.

The ER can be expressed as

$$ \frac{D}{\mu } = \left( {\frac{{n_{0} }}{e}} \right)\left[ {\frac{{\partial n_{0} }}{{\partial (E_{F} - Z)}}} \right]^{ - 1} $$
(2.4)

where \(Z\) is given by

$$ A_{11} (Z,\eta_{g} ,n_{y} ) = 0 $$
(2.5)

Thus by using (2.2)–(2.4), we can study the DMR numerically.

Incidentally, the photo current \(I\) can be written as

$$ I = \frac{{\alpha_{0} eg_{v} k_{{\text{B}}} T}}{\pi \hbar }\sum\limits_{{n_{y} = 1}}^{{n_{{y_{\max } }} }} {\ln (1 + \exp [(E_{F} - (Z + W - h\upsilon ))(k_{{\text{B}}} T)^{ - 1} ]} $$
(2.6)

where \(\alpha_{0}\) is the probability of photoemission

The field emitted current (\(i_{{\text{f}}}\)) assumes the form

$$ I = \frac{{2eg_{v} k_{B} T}}{h}\sum\limits_{{n_{y} = 1}}^{{n_{{y_{\max } }} }} {} [\ln (1 + \exp [(E_{F} - Z)(k_{{\text{B}}} T)^{ - 1} ]\exp ( - Q)] $$
(2.7)

where

$$ Q = \frac{{4[A_{11} (V_{0} ,\eta_{g} ,n_{y} )]^{3/2} }}{{3eF_{x} \left[ {A^{\prime}_{11} (V_{0} ,\eta_{g} ,n_{y} )} \right]}},\;V_{0} = E_{F} + \phi_{w} $$

2.3 Results and Discussion

The plot of the normalized EEM in HDNWs of MOS2 versus n0 for three sub-bands is given in Fig. 2.1. The plots of normalized DMR (\(\overline{D}\)) in HDNWs of MOS2 versus n0 are given in Figs. 2.2, and 2.3 shows the same for HDNWs of \({\text{InSb}}\) for the purpose of relative comparison. Figures 2.4 and 2.5 explore the normalized photo current (\(\overline{I}\)) from HDNWs of MOS2 versus \(n_{0}\) and the same for HDNWs of \({\text{InSb}}\) respectively. Figures 2.6 and 2.7 exhibit the plots of the normalized FE and TP for different HDNW biomaterials versus n0 respectively.

Fig. 2.1
A graph plots the normalized E E M for n subscript y equals 1, 2, and 3 with respect to n subscript 0. The three curves trend in increasing order.

Plot of the normalized EEM versus n0 for three different values of ny in HDNW MOS2 where dy = 20 nm

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Fig. 2.2
A graph plots the normalized D M R for d subscript y equals 10, 20, 30, and 40 n m with respect to n subscript 0. The three curves trend in increasing and decreasing order.

Plot of the \(\overline{D}\) in HDNWs of MOS2 versus n0 for four different values \(d_{y}\) as shown in the figure

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Fig. 2.3
A graph of normalized 1 D D M R versus n subscript 0 for d subscript y equals 10 and 20 n m. The two lines start off constant, then rise, and finally remain constant.

Plot of the \(\overline{D}\) versus \(n_{0}\) for the NWs of \(n - {\text{InSb}}\) with two different values of \(d_{y}\)

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Fig. 2.4
A graph plots normalized photocurrent for film thicknesses of 10, 20, 30, and 40 n m with respect to normalized n subscript 0. The four curves trend in increasing and decreasing order.

Plot of the \(\overline{I}\) from NWs of MOS2 versus n0 for four different values of film thickness

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Fig. 2.5
A graph plots normalized photocurrent with respect to normalized n subscript 0. The curve trends in increasing order.

Plot of the \(\overline{I}\) versus \(n_{0}\) for the NWs of \(n - {\text{InSb}}\)

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Fig. 2.6
A graph plots the normalized field emitted current with respect to the electric field for three different H D N W biomaterials. The three curves trend in increasing and decreasing order.

Plot of the normalized field emitted current versus electric field for three different HDNWs of biomaterials

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Fig. 2.7
A graph plots the normalized F E and T P with respect to b subscript 0 for four different H D N W biomaterials. The four curves trend in decreasing and increasing order.

Plot of the normalized TP versus n0 for four different HDNWs of biomaterials as shown by a, b, c and d, respectively

Full size image

The salient features are given below:

  1. 1.

    In Fig. 2.1, the EEM increases with increasing n0 where the value of EEM for \(n_{y} = 1\) is the greatest.

  2. 2.

    In Fig. 2.2, the \(\overline{D}\) in HDNWs of MOS2 oscillates with enhanced \(n_{0}\), and the magnitude and nature of oscillations are totally different as compared with the \(\overline{D}\) in HDNWs of other material as given in Fig. 2.3. The quantum signatures of two different types of 1D motion can be assessed by comparing Figs. 2.2 and 2.3.

  3. 3.

    From Figs. 2.4 and 2.5, it appears that the \(\overline{I}\) HDNWs of MOS2 oscillates with \(n_{0}\) in radically different manner as compared with that from HDNWs of other materials.

  4. 4.

    From Fig. 2.6, we note that the field emitted current oscillates with increase in electric field due to Van Hove singularities

  5. 5.

    From Fig. 2.7, we note that the TP increases with increasing \(n_{0}\) in oscillatory ways.

Most important to realize is that the quantum signatures in all the cases are not only totally different, but also the variations of the said electronic quantities as compared with that of HDNWs different materials excluding biocompounds are also different.

2.4 Conclusion

In this chapter, we study the EEM, ER, EP, FE and the TP in heavily doped nanowires (HDNWs) of different biosensing materials together with the relative comparison of the said transport features with that of the HDNW compounds. We observe that the EEM is quantum number dependent. The ER oscillates with the electron statistics (\(n_{0}\)), and the magnitude and nature of oscillations are totally different as compared with the ER in HDNWs of other materials talking HDNW of InSb as an example. The Einstein’s photo current from HDNWs of different biosensing materials also oscillates with \(n_{0}\) in radically different fashion as found from HDNWs of other materials. The field emitted current oscillates with increase in electric field due to Van Hove singularities, and the TP increases with increasing \(n_{0}\) in oscillatory ways. The most important realization is that the quantum signatures in all the cases are not only totally different, but also the variations of the said electronic quantities as compared with that of HDNWs different compounds excluding biomaterials are also different.